FORMATION OF THE IMPULSE RESPONSE OF A RECURSIVE LOW-PASS FILTER WITH FINITE IMPULSE RESPONSE AS A SUM OF QUASI-HARMONICS OF A TRUNCATED FOURIER SERIES
Abstract
The problem of reducing the number of arithmetic operations in digital filtering algorithms is highly relevant, as it directly impacts power consumption, processing speed, and hardware costs. Under strict power efficiency requirements for mobile and embedded systems, minimizing multiplication and addition operations becomes a critical design factor. This paper presents a method for implementing a recursive filter with a finite impulse response (FIR) based on a truncated sinc function smoothed by a window (weighting function), represented as a sum of quasi-harmonic functions. These quasi-harmonic functions with different frequencies are polynomials of degree r. The study adopts a second-degree polynomial as a baseline and proposes a numerical method for increasing the polynomial order to improve the accuracy of the approximation. Accuracy analysis demonstrates that using 4th- and 6th-order polynomials achieves an approximation error of less than 1%. The coefficients of the non-recursive part of the filter are computed via inverse finite differences of the original FIR impulse response. These coefficients are integers whose values depend on the number of samples (length) of the half-period of the quasi-sinusoidal function, simplifying the implementation of such a recursive FIR (RFIR) filter on a field-programmable gate array (FPGA). Numerical analysis of finite differences for each quasi-sinusoid revealed that quadratic approximation requires only 16 samples but results in relatively high side-lobe levels (–30 dB). Switching to 4th-order approximation increases the number of non-zero coefficients to 20 and significantly reduces (by 13 dB) the stopband magnitude of the frequency response, reaching –43 dB.
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